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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 18v, page id 00068196:0036
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vr Co de +A dco + (n: a) co dZ + Z dZ = 0 sic transformare liceat
in ipsam edo + β dv + (V: a) v dx + B dx = =0 ut fial D = β7: a
fiat7 = x + h et co= v. fietque
V dv + A dv + (M: a) v dx + E dx = 0 et fieri dabet s p. at MM: a) dh _ = 3 ( 
+ (n: a) D dhdx + ex d h dx
seu RM: a (1+ dh) = E (1+ d E) 
Sed ita rursus evanesinit assumtitia h, unde difficile videtur obtinere
Quaesitum
hyy + my +n =0 inde rhydy + mdy + yydh + ydm + dn
sit aequalio iuta ejusdemformae b)du + cdy + eyydx + yfdx + _gdx,
fiet: h= Eb M = C n = sgdx et fit b = sedx et c = sfdx
fe  h yB i m - o fiet hydu h dy + ydh + am = 0, et
5yyy 3K yydy +7dn M 2m ydy + ndy +. yy dntydn + d
BKy + Myy + ny + p =0
Sit equatio data  ydy + β dys +), dx + B dx =0 quae multiplicetur per 3k7 + h.
et fiet 3k gy dy + 3K Bydy + 3HYgydx t 3KBYdx +h βdy + 3K Jyy ax 13kBd dx + h8t
BC
k est constans
D=
+ h julus.
fiet 2m = 3RB + h et n = hβ et p = shBdx
+hVy ax
dm = 3kV dx Ergo 2K dβ + = dh = 3K) dx adeoque h = 3 3K (β3 - 3K s)dx
dn = 3H x 3k D dx + h) dx = βdh + h d β. sed ita prodit ipsius d d
hin aliquid generale ducere
quid si sie  gy + my +- fiet rydy + ndy + ydm + dn = 0 Haec
Clhty dys +my
valor ad huc semel Non ergo
yamyu +nytp -o fiet
equatio multiplieltur per k et fiet 2Ky dus Hm du + Kyam + Han = 0
rurus sit aequatio vata yndys + Bdy+Vgdxdx=o haec muttiplicetur per 2ky + h
Ex differe sit y+ myyt ang +aap =0 di hoc differentiata dabit:
3yydy +2 mydy + andy + yydm + aydn + aa dp=o  haec multiplicetur per K
i
tiet 3kyy dys + 2Kmyay + aKn dy + kyydm + aKy du + aa Kd p = 0. it data
solvenda ydy + β dy + (3: a) y dx + d dx = haec multiplicetur per 3Ky + ah, et
fiet (56) 0 = 3Kyy dy + ah ydy + a (h dy + 3V: a) K yydx + Vhydx + a hddx -0
+ 38K 7 dx
Hinc compatando  exydy fit ah aequationes 3 et S. setrat
+3BKydy
ic
ex ydy, ah = 2Km 4 3BK et ex dy fit (7 nod βh = kn. seu ex essit 2(  -3 (BK a n
et ex yydx sit Kam = 3V: a) K dX seu dm  = (9no) 3(V: a) K dx seu m = (3: a) S) x
et ex ydx fit, fit a si dn = (10mo) Vh dx + 3DKdx et po h ponendo valorem ex aeg. 7.
fiet a edu = V (N: B) K dx + 3bR dx. sed ex aqu. 8 est 2βB: a) sax - 3 33  an. sed ita
 et ex 12 et
cum _ evanescat, natet B haberi bis, adeoque non haberi arbitrarias sufficiea
1 fit  dn    fy) 2. 3)) sp ax 2aY βd∞ + 3 aa Bdx = 2.3
Redeamus ad simplicia ibi mi Miscendo lagarithinum aby, sit
m at fdy: y) + my +an =o et differltiando fit al dy: y + m dus + yam + adm =0
seu at myd ygem mydu + al dus + yydm + y du et posito  m esse constantum
fiet mydyus + al dy +y dn =0 Hinc sipody et du multiplicata per eandem varia.
vilem, potest solvi aequatio hujus formae, sed sed haec aliunde solvi potest, seu t
ponamd, l esse variabulem riet aleus: y) + a sag: y dl + mdus + adu = 0 et pro sdy: Y subtitu
endo valorem ex aequ. pro po sita siet all dus : y) = m al - an dl +lmdus + aldu et fiet
Ry dy tiaß dy + Xyy dx + aBy dx = quan multiplicando per : m: aa. fiet
Im ydy + In Bdy ut  ImV: aa) yyax + Lnif. ) dy dx Im(B: a) y dx = 0 fiet al = m (3
senique n AV. a nV: a) ax + asdn = lmB: a) dx
indx + aa Ban = (BCm Ax =) ABI dx
Ydx 13 x d β3+ 2. 3 aa B3 d B
anal y = 0 sit aequatio hujus formae
aH dy me jus dF.= lmydys + allius= myy dl + al da
eu
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